Motion Planning of a Water Tank with Differential Flatness ...
Transcript of Motion Planning of a Water Tank with Differential Flatness ...
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Motion Planning of a Water Tank with Differential
Flatness and Optimal Control
Dimitar Ho
Brian Grunloh
Nick Jenson
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Problem Definition
• Project Goal
– Move a tank containing a fluid to another location within a given time with minimal sloshing
• Potential Project Applications
– Industrial process control moving tanks of liquid from station to station
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Example of Constant Velocity
• Container traveling 1.5 meters in 2 seconds
• All simulations run with 0.3 x 0.3 meter container that is half filled
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Proposed Solution Methods
• Differential Flatness
– Based on Conservation of Mass and Momentum Equations
– Reduction to form of 1-D wave equation
• Optimal Control of Equivalent ODE Approach
– Pendulum Approximation
– Double Pendulum Approximation
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Differential Flatness: Assumptions
• Saint Venant Equations (Shallow Water) – Implies vertical velocities are much smaller than horizontal
velocities and are therefore negligible.
• 1-Dimensional Motion – Restricted to horizontal translation (non-rotational)
• Neglecting Coriolis, frictional, and viscous forces • Flat bottom, rectangular fluid container • Motion begins and ends at steady-state
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Differential Flatness: Governing Equations
• General Results from the Conservation of Mass and Momentum Equations
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Differential Flatness: Results
• y(t) becomes arbitrary function between initial and final states – For minimal sloshing at steady state condition, must
choose y(t) to have first and second derivatives equal zero at boundaries
• System is steady state controllable if initial state is zero – Can be steered from a steady-state position to any other
steady position
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Simulations – Differential Flatness
• Select y(t) to be a linear function
• Constant velocity model under equivalent time and distance
Model simulated to travel 1.5 meters over 2.0 seconds
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Simulations – Differential Flatness
• Select y(t) to be a modified cosine function
• Constant velocity model under equivalent time and distance
Model simulated to travel 1.5 meters over 2.0 seconds
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Equivalent ODE
• Approach:
– Represent the water tank as a pendulum ODE system
– Calculating steering, D(t), with Optimal Control using the approximate ODE model
• Assumptions:
– Inviscid fluid
– Irrotational fluid
– Incompressible fluid
– No surface tension
D
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Equivalent ODE Approach: Modeling of the system
Simplification of the PDE to an ODE model consisting of N-pendulums, representing the first N eigenmodes of the water tank
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Equivalent ODE Approach: Optimal Control
Formulating the minimum sloshing motion planning problem as an Optimal Control Problem using the
equivalent pendulum model
Optimal Control Problem
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Equivalent ODE Approach: Solution of the Optimal Control Problem
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Simulations – Equivalent ODE
• Singular Pendulum Approach
• Constant velocity model under equivalent time and distance
Model simulated to travel 1.5 meters over 2.0 seconds
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Equivalent ODE Approach: Adding Tilting Motion
Adding a tilt-DoF enables us to model the water tank system as a double pendulum.
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Simulations – Equivalent ODE
• Double Pendulum Approach with Rotation
• Constant velocity model under equivalent time and distance
Model simulated to travel 1.5 meters over 2.0 seconds
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Method Comparisons
Constant Velocity
Flatness with cosine y(t)
Single Pendulum
Double Pendulum
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Conclusions
• Both Differential Flatness and Optimal Equivalent ODE provide enhanced trajectories for minimizing sloshing
• Proper choice of y(t) improves trajectory
• Double Pendulum approach is optimal for reduction of fluid travel up the sides of the container due to container rotation
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Future Steps
• Verification of simulation results
– Implementation of resulting controls on linear stage system in Hesse Hall Lab